A particularly interesting class of nonlinear projections are the pinned distance functions \(d_p(q) = |p-q|\) . In [1]}, Orponen proved a packing dimension version of the Falconer distance conjecture in the plane for Ahlfors-regular sets by showing that the nonlinear projection \(d_p(q)\)  could be analyzed using multi-scale averages of suitably chosen linear projections, and thus the problem of computing the size of the pinned distance set \(\Delta _p(E)=\lbrace d_p(q)\colon q\in E\rbrace \)  was closely related to the problem of computing the size of certain orthogonal projections of \(E\)  at various scales and locations. This powerful new idea has been extended in subsequent works [2]}, [3]}, [4]}. In particular, Shmerkin [4]} proved an analogue of Theorem REF  for pinned distances.
